The Tangent Line to a Curve at a Point Closely

question 145

Multiple Choice

The tangent line to a curve at a point closely approximates the curve near the point. In fact, for x-values close enough to the point of tangency, the function and its tangent line are virtually indistinguishable. This problem explores this relationship. Write the equation of the tangent line to the curve at the indicated point. To check your answer, use a graphing utility to graph both the function and its tangent line, and repeatedly zoom in on the point of tangency until the function and the tangent line cannot be distinguished. ​ ​

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Definitions:

Heavy Drinking

The consumption of large quantities of alcohol in a short period, often leading to health risks and social problems.

Sleep Disturbance

Disruptions in the normal sleep cycle, which can include difficulties in falling asleep, staying asleep, or experiencing restful sleep.

Depression

An emotional state marked especially by sadness, inactivity, difficulty in thinking and concentration, a significant increase or decrease in appetite and time spent sleeping, feelings of dejection and hopelessness, and sometimes suicidal tendencies.

Suicide Risk

The possibility that an individual will attempt to take their own life or engage in self-harmful behaviors, often assessed through various psychological evaluations.